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Quality-Dependent Stochastic Networks:Is FIFO Always Better Than LIFO?

Yair Liberman∗

libyair@gmail.com

Department of Statistics and Operations Research,

School of Mathematical Sciences

Tel Aviv University, Israel

Uri Yechiali∗

uriy@tauex.tau.ac.i

Department of Statistics and Operations Research,

School of Mathematical Sciences

Tel Aviv University, Israel

ABSTRACTIn various service and production systems, such as metal process-

ing, fresh food industry, polymer forming, etc., the quality of the

final product is a function of its total sojourn time in the system.

We consider (i) an M/G/1-type single-server queue, as well as

(ii) open tandem Jackson networks, and analyze the quality of

products traversing through each system under two service dis-

ciplines â FIFO and LIFO. Although the mean sojourn times in

anM/G/1 queue are equal under both service disciplines (that is,

E[WLI FO ] = E[WF I FO ]), the variance ofWLI FO is larger than the

variance ofWF I FO . We show that, when quality is the measure of

performance, FIFO is not necessarily better than LIFO. We consider

several service-time distributions and show under what values of

the parameters one discipline is better (or worse) than the other.

In particular, the mean quality under LIFO is better than the mean

quality under FIFO for all values of the traffic intensity. Moreover,

the mean quality under the FIFO service regime drops sharply to

0 when the traffic intensity approaches 1, while the mean quality

under LIFO is bounded below. Numerical results as functions of the

system parameters are presented and discussed for both theM/G/1queue and the tandem Jackson network.

CCS CONCEPTS•Mathematics of computing→Queueing theory; •Networks→ Network performance evaluation.

KEYWORDSQuality, FIFO, LIFOM/G/1,Tandem Jackson Network

ACM Reference Format:Yair Liberman and Uri Yechiali. 2019. Quality-Dependent Stochastic Net-

works: Is FIFO Always Better Than LIFO?. In Proceedings of ValueTools ’20:13th EAI International Conference on Performance Evaluation Methodolo-gies and Tools (ValueTools ’20). ACM, New York, NY, USA, 8 pages. https:

//doi.org/10.1145/1122445.1122456

∗Both authors contributed equally to this research.

Unpublished working draft. Not for distribution.Permission to make digital or hard copies of all or part of this work for personal or

classroom use is granted without fee provided that copies are not made or distributed

for profit or commercial advantage and that copies bear this notice and the full citation

on the first page. Copyrights for components of this work owned by others than ACM

must be honored. Abstracting with credit is permitted. To copy otherwise, or republish,

to post on servers or to redistribute to lists, requires prior specific permission and/or a

fee. Request permissions from permissions@acm.org.

ValueTools ’20, May 18-20, 2020, Tsukuba, Japan© 2019 Association for Computing Machinery.

ACM ISBN 978-1-4503-9999-9/18/06. . . $15.00

https://doi.org/10.1145/1122445.1122456

1 INTRODUCTIONQueueing theory deals primarily with probabilistic analyses of

various types of service systems under a variety of probabilistic

assumptions on the processes driving such systems. Usually the

study is directed towards analyzing the distribution of the number

of customers in the system and of their sojourn times, as well as

finding the corresponding means and variances. In many produc-

tion systems, e.g. fresh food industry, metal processing, polymer

forming, etc., the quality of products traversing through the system

deteriorates along with their sojourn time (see e.g. Liberopoulos et

al, 2007 [18]). In such cases it is essential to determine the order of

service of products, namely, serving them according to the First In

First Out (FIFO) regime, or alternatively according to the Last In

Last Out (LIFO) regime (Other services regimes, such as Random

Order Of Service or Processor Sharing are not discussed in the

current study). Also, it is important to determine the best order in

which the product passes through a network of service sites, and

as a result improve the product’s quality and thus its value and

reputation (see, e.g., a survey by Inman et al., 2013 [9]).

Deterioration of products is treated extensively in the manufac-

turing and inventory management literature, and it often aims at

calculating the level of inventory stock and necessary lead time in

purpose of minimizing costs and avoiding shortage (see e.g. Van

Horenbeek et al., 2013 [25]). The inventory’s quality analysis mainly

deals with defective items due to imperfect manufacturing process,

or damages (see e,g. see Kim and Gershwin, 2005 [14]). In addition,

there is an abundance of research over quality-dependent prod-

ucts, where the majority of works deal with perishable products,

with quality defined as a time-based dichotomous function (see e.g.,

Cooper, 2001 [6]; Berk and Gurler, 2008 [2]; Liberopoulos et al., 2007

[19]; Avinadav and Arponen, 2009 [1]). Naebulharam and Zhang

(2013) [20] studied a Bernoulli quality model which determines the

quality of each part, defective or non-defective, by a series of i.i.d.

Bernoulli random variables. Coledani et al. (2015) [5] considered a

discrete-time discrete-state Markov chain to examine the effects of

buffer sizes on the probability that a product is defective or not in a

multi-stage production system. Bortolini et al. (2016) [3] described

a piecewise linear function between quality and time, and used it to

estimate the market purchase probability. Goyal and Giri (2001) [8]

and Perlman and Yechiali (2019) [21] assumed an exponential decay

function of the product’s quality. Our model also follows the expo-

nential decay assumption and quantitatively defines the quality of

a product passing through a production (or service) system.

The sojourn time of a product depends on various system char-

acteristics, such as job-arrival process, distribution of service (pro-

cessing) times, inner order of service and the sequence of service

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stations in case of a tandem network. Studies on manufacturing

systems examined the effect of system parameters on productivity

efficiency and production costs. Regarding the single server model,

Levi and Yechiali (1975) [16] indicated the decomposition phenom-

enon of waiting times for the M/G/1 queue with multiple vaca-

tion, and derived the corresponding Laplace-Stieltjes Transform

(LST). Scholl and Kleinrock [23] investigated the multiple vacation

M/G/1 queue and derived the LSTs of waiting times under the FIFO,

LIFO and ROS (Random Order of Service) regimes. Rosenberg and

Yechiali (1993) [22] studied the batch arrivalMx /G/1 queue withsingle and with multiple vacations and derived explicit formulas

for the LSTs, means and second moments of the waiting time for

both FIFO and LIFO service-order regimes. They showed that al-

though E[WF I FO ] = E[WLI FO ], the second moment under LIFO

is larger than that under the FIFO, and gave an explicit formula,

E[(W

qF I FO )

2]= E

[(W

qLI FO )

2]/(1 − ρ), for all cases. The current

study relies on those results to compute the means and variances

of a product’s quality for theM/G/1 queue under both LIFO and

FIFO service regimes.

Research over a network of queues has began with the seminal

works of R.R.P. Jackson (1954 [12]; 1956 [13]) and J.R Jackson (1957

[10]; 1963 [11]). They developed a Markovian model, called Jackson

network, and characterized the transition probabilities and basic

system parameters of such networks. In a further work by Dallery

and Frein (1993) [7] a manufacturing system was investigated as an

open tandem Jackson network with either finite or infinite buffer

between sites. They showed how using the decomposition phenom-

ena allows the computation of basic system performance measures.

Other researches (see Yechiali, 1988 [27]; Brandon and Yechiali, 1991

[4]) studied the n-node tandem queueing network with Bernoulli

feedback to the first node and derived the LST of the total sojourn

time of a product in the system. Perlman and Yechiali (2019) [21]

investigated an n-site tandem network where each site consists of

two stages: a processing stage and an inspection stage. The pro-

cessing operation is a generally distributed random duration which

either does or does not conclude successfully; in the latter case,

the operation is repeated immediately. Once the processing stage

concludes successfully, the product goes through an inspection

stage which determines whether it will (i) require an additional

processing and move forward to the next station; or (ii) it is found

‘good’ and exits the system with quality value depending on its

sojourn time; or (iii) it is declared as ‘failed’and exits the system

with zero quality value. In their study they derived explicit results

for the mean sojourn time of a product in the system, and its mean

quality while assuming that each site is comprised of twoM/M/1-

type independent queues with FIFO service regime. In our work

we expand this model and give an explicit formula for a tandem

Jackson network where each site is an M/M/1 queue with LIFO

service regime, and compare the results between the FIFO and the

LIFO disciplines.

This paper treats two basic models (i) an M/G/1-type single-

server queue, and (ii) an open tandem Jackson network. The objec-

tive is to analyze the quality of products traversing through each

system under two service disciplines — FIFO and LIFO. In Section

2 theM/G/1 model is presented, and formulas are derived for the

quality’s mean and variance for both FIFO and LIFO service regimes.

Two service time distributions are investigated: Exponential, and

two-phase Erlang. The means and variances under each service

time distribution are compared. It is shown that FIFO is not neces-

sarily better then LIFO, as probably expected, but rather, the mean

quality under LIFO is better then the mean quality under FIFO for

all values of the traffic intensity. In Section 3 the tandem Jackson

network is analyzed and formulas are derived for the mean quality

of a product passing through the system, under both the FIFO and

LIFO service regimes. Numerical results further show the relations

between the two service disciplines. Conclusion are discussed in

Section 4.

2 MODEL FORMULATION - SINGLE SERVERWe consider an M/G/1 queueing system where a Poisson stream of

jobs (customers, calls, particles, products, etc.), with rate λ, flowsinto a single-server station. In steady state, let L denote the number

of jobs in the system, and let Pn = P(L = n), n = 0, 1, .... Service

time, B, of an arbitrary job is generally distributed with finite mean

E[B], finite second moment E[B2], and Laplace Stieltjes Transform

(LST) B(s). LetW qdenote the queueing time (not including service)

of a job, and letW =W q + B denote its overall sojourn time. Let

ρ = λE[B] < 1.

Consider the FIFO (First In First Out) service regime. The LST

of the queueing time of an arbitrary job is given by (see Kleinrock

(1975) [15], p. 199)

W q F I FO (s) =s(1 − ρ)

[s − λ + λB(s)](1)

which is known as the Pollaczek—Khinchine formula. The mean

queueing time, E[WqF I FO ], is given by

E[WqF I FO ] =

λE[B2]

2(1 − ρ). (2)

The LST of the sojourn time of a job is given by

WF I FO (s) = W q F I FO (s)B(s) =s(1 − ρ)B(s)

[s − λ + λB(s)]. (3)

Let θ denote the length of a busy period, which is the time from

the first arrival to an empty system until the system is empty again

for the first time. The mean and LST of θ are the solution for the

following equations (see Kleinrock (1975) [15], p. 212): E[θ ] = E[B]1−ρ ,

and

θ (s) = B[s + λ − λθ (s)]. (4)

Now, consider the non-preemptive LIFO (Last In Fast Out) service

discipline where the server serves each job completely, and imme-

diately afterwards begins to serve the last job to have arrived (if

any). The LST and mean of the queueing time of an arbitrary job

are given (see Rosenberg and Yechiali 1993 [22]) by

W qLI FO (s) = (1 − ρ) +λ[1 − θ (s)

]s + λ

[1 − θ (s)

] , (5)

and

E[WqLI FO ] =

λE[B2]

2(1 − ρ)= E[W

qF I FO ]. (6)

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The LST of the sojourn timeWLI FO is given by

WLI FO (s) = WqLI FO (s)B(s) = (1 − ρ)B(s) +

λ[1 − θ (s)

]B(s)

s + λ[1 − θ (s)

] . (7)

It follows (see Rosenberg and Yechiali (1993)[22]), that although

E[WqF I FO ] = E[W

qLI FO ], the second moments are related by

E[(WqLI FO )

2] = 1

1−ρ E[(WqF I FO )

2].

2.1 Quality analysis forM/G/1 queueWhile waiting in line, a product suffers degradation in its quality.

The quality of a product that sojournsW units of time is considered

here as Q = δ e−γW , where δ is the potential quality in caseW =0, and γ is a positive constant serving as a scaling factor. This

formulation follows other works dealing with product degradation

(see, e.g. Perlaman and Yachiali (2019) [21] and references there).

For the M/G/1 queue, the expected value of a product’s quality

under the FIFO service regime is given by (see Eq. (3))

E[QF I FO ] = E[δe−γWF I FO ] = δWF I FO (γ ) = δγ (1 − ρ)B(γ )

γ − λ + λB(γ ). (8)

Clearly, when ρ → 1, E[WF I FO ] → ∞ and E[QF I FO ] → 0. How-

ever, when λ → 0 (ρ → 0), E[QF I FO ] → δ B(γ ). Indeed, inthe latter case, upon arrival, every product finds an empty sys-

tem so that its sojourn time is only its service duration B. Thus,

E[QF I FO ] = E[−δe−γ B ] = δ B(γ ).The second moment ofQF I FO can be calculated by using Eq. (8):

E[Q2

F I FO ] = E[δ2e−2γWF I FO ] =δ2WF I FO (2γ ) =

δ22γ (1 − ρ)B(2γ )

2γ − λ + λB(2γ ).

(9)

The Variance of QF I FO is calculated by

V [QF I FO ] =E[Q2

F I FO ] − E2[QF I FO ] =

δ22γ (1 − ρ)B(2γ )

2γ − λ + λB(2γ )−

[δγ (1 − ρ)B(γ )

γ − λ + λB(γ )

]2

.(10)

Now, when ρ → 1, V [QF I FO ] → 0, while when λ → 0 (ρ → 0),

V [QF I FO ] → δ2B(2γ ) − δ2B2(γ ). (11)

Regarding the LIFO service discipline, by using Eq. (7) the mean

quality of a product is calculated as

E[QLI FO ] =δE[e−γWLI FO ] = δWLI FO (γ ) =

δ

[(1 − ρ)B(γ ) +

λ[1 − θ (γ )]B(γ )

γ + λ[1 − θ (γ )

] ] . (12)

In this case too, limλ→0E[QLI FO ] = δ B(γ ). However, when ρ → 1

(i.e. λ → 1

E[B] ),

lim

ρ→1

E[QLI FO ] =[1 − θ (γ )]B(γ )

E[B]γ + [1 − θ (γ )]. (13)

The second moment of QLI FO is calculated as

E[Q2

LI FO ] =E[δ2e−2γWLI FO ] = δ2E[e−2γWLI FO ] = δ2WLI FO (2γ )

= δ2(1 − ρ)B(2γ ) + δ2λ[1 − θ (2γ )B(2γ )

2γ + λ[1 − θ (2γ )].

(14)

The Variance of QLI FO can be computed by using Eq.(12) and (14):

V [QLI FO ] =E[Q2

LI FO ] − E2[QLI FO ] =

δ2(1 − ρ)B(2γ ) + δ2λ[1 − θ (2γ )]B(2γ )

2γ + λ[1 − θ (2γ )]−

δ2[(1 − ρ)B(γ ) +

λ[1 − θ (γ )]B(γ )

γ + λ[1 − θ (γ )

] ]2.(15)

For the LIFO case, when ρ → 1 (i.e. λ → 1

E[B] ),

lim

ρ→1

V [QLI FO ] =

δ2 lim

ρ→1

[[1 − θ (2γ )]B(2γ )

E[B]2γ + [1 − θ (2γ )]−

([1 − θ (γ )]B(γ )

E[B]γ + [1 − θ (γ )]

)2].

(16)

For λ → 0, as in the FIFO case, from Eq. (15),

V [QLI FO ] → δ2B(2γ ) − δ2B2(γ ). (17)

In Sections 2.2, and 2.3 we consider two service distributions,

namely, (i) Exponential and (ii) two-phase Erlang.

2.2 Quality analysis for Exponential servicetimes

2.2.1 FIFO Service Regime. For theM(λ)/M(µ)/1 queue, where

B ∼ Exp(µ) and B(s) =µ

µ+s , the sojourn time of a product under

the FIFO service regime,WF I FO , is distributed Exponentially with

parameter µ − λ. Thus,

WF I FO (s) =(µ − λ)

(µ − λ) + s. (18)

Then, the expected quality of a product is given by Eqs. (8) and

(18):

E[QF I FO ] = δWF I FO (γ ) = δ(µ − λ)

(µ − λ) + γ. (19)

As in Section 2.1, when λ → 0, E[QF I FO ] → δµ

µ+γ = δ B(γ ),

whereas when ρ → 1 (i.e. λ → µ), E[QF I FO ] → 0.

The Variance of QF I FO is calculated by using Eq. (10):

V [QF I FO ] = δ2(µ − λ)

[1

µ + 2γ − λ−

µ − λ

(µ + γ − λ)2

]. (20)

It follows that, as in Section 2.1, when λ → µ (i.e. ρ → 1),V [QF I FO ] →

0 (see also Fig. 1), while when λ → 0,

V [QF I FO ] → δ2µ

[1

µ + 2γ−

µ

(µ + γ )2

]= δ2

[B(2γ ) − B(γ )2

]. (21)

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2.2.2 LIFO Service Regime. For the LIFO service discipline in an

M(λ)/M(µ)/1 queue, where B(γ ) =µ

µ+γ , we get by Eq. (12):

E[QLI FO ] =δ (µ − λ)

µ + γ+

δλµ[1 − θ (γ )]

(µ + γ )(γ + λ[1 − θ (γ )]). (22)

Note that the length of the busy period in a work conserving system

is independent of the service regime. Thus, by using Eq. (4), we

obtain

θ (γ ) = B(γ + λ − λθ (γ )) =µ

µ + [γ + λ − λθ (γ )],

(23)

resulting in

θ (γ ) =γ + λ + µ −

√(γ + λ + µ)2 − 4λµ

2λ. (24)

By substituting Eq. (24) in Eq. (22) we get an explicit expression for

E[QLI FO ]. That is,

E[QLI FO ] =

δ

µ + γ

[2µ − λ −

2γ µ

λ + γ − µ +√(γ + λ + µ)2 − 4λµ

].

(25)

Indeed, as in Section 2.1, when λ → 0, E[QLI FO ] = δµ

µ+γ =

δ B(γ ), whereas when ρ → 1 (i.e. λ → µ)

E[QLI FO ] →δ

µ + γ

[µ −

2γ µ

γ +√γ 2 + 4γ µ

]. (26)

The variance of QLI FO , when substituting B(2γ ) =µ

µ+2γ in Eq.

(15), results in

V [QLI FO ] = δ2(µ − λ)

(µ + 2γ )+ δ2

λµ[1 − θ (2γ )]

(µ + 2γ )(2γ + λ[1 − θ (2γ )])

−δ2

(µ + γ )2

[(µ − λ)2 +

2λµ(µ − λ)[1 − θ (γ )]

γ + λ[1 − θ (γ )]+[

λµ[1 − θ (γ )]

γ + λ[1 − θ (γ )]

]2],

(27)

where

θ (2γ ) =2γ + λ + µ −

√(2γ + λ + µ)2 − 4λµ

2λ. (28)

2.2.3 Comparison between LIFO and FIFO forM(λ)/M(µ)/1queue. In this section the mean qualities and quality variances

under the LIFO and under the FIFO service regimes are compared

for theM/M/1 queue. Using Eqs. (19) and (25), the ratio between

the two mean qualities is given by

E[QLI FO ]

E[QF I FO ]=

(µ − λ + γ )

(µ − λ)(µ + γ )

[2µ − λ −

2γ µ

λ + γ − µ +√(γ + λ + µ)2 − 4λµ

].(29)

Propositions 1 and 2 below show the advantage of the LIFO over

the FIFO with respect to mean qualities.

Proposition 2.1. 1 For µ,γ > 0 and 0 < λ ≤ µ,

limλ→µE[QLI FO ]

E[QF I FO ]= ∞.

Proof. The claim follows directly from Eq. (29). □

Proposition 2.2. 2 For γ > 0 and 0 < λ < µ,E[QLI FO ]

E[QF I FO ]> 1.

Proof. By Eq. (29)E[QLI FO ]

E[QF I FO ]> 1 ⇐⇒

(µ − λ + γ )(2µ − λ)[λ + γ − µ +

√(γ + λ + µ)2 − 4λµ

]+ 2γ µ >

(µ − λ)(µ + γ )[λ + γ − µ +

√(γ + λ + µ)2 − 4λµ

].

(30)

LetC B λ+γ−µ+√(γ + λ + µ)2 − 4λµ denote the the numerator

of θ (γ ) in Eq. (24). Clearly,C > 0. SubstitutingC in Eq. (30) translates

to

(µ − λ)(2µ − λ)C + γ (2µ − λ)C + 2γ µ > (µ − λ)(µ + γ )C ⇐⇒

(µ − λ)2C + γ µC + 2γ µ > 0,

(31)

which holds since γ µ > 0 and (µ − λ)2 > 0. □

Figure 1 depicts the behaviour of the mean qualities (Figure 1(a))

and variances (Figure 1(b)) for both FIFO (red) and LIFO (blue)

service regimes as functions of λ when µ = 20, γ = 1, δ = 1.

E[QLI FO ], is a monotonously decreasing concave function of λ,approaching 0.76 as λ → µ (see Eq. (26)). E[QF I FO ] is also concave,

but in contrast to E[QLI FO ], it decreases sharply when λ gets closerto µ, reaching 0 as λ = µ. Figure 1(a) illustrates graphically the

claims of proposition 1 and 2.

(a) E[Q ] as a function of λ (b) V [Q ] as a function of λ

Figure 1: Mean and variance of a product’s quality under FIFO andunder LIFO service regimes for Exponential service times as a

function of λ, when µ = 20, γ = 1, δ = 1

Figure 1(b) depicts the behaviour of the variances of the two

service regimes as functions of λ for the same parameter values as in

figure 1(a). It is seen that both variances increase when λ is small or

moderate. As long as λ is smaller then 18.2,V (QF I FO ) < V (QLI FO ),

but when λ exceeds 18.2, the ratio turns over and V (QF I FO ) >

V (QLI FO ). As λ further increasesV (QF I FO ) changes its direction at

its highest valueV (QF I FO ) = 0.09, and then drops sharply towards

0, crossing V (QLI FO ) at λ = 19.6.

The above observations are explained as follows: When λ is

small, there is a significant probability under the LIFO regime that

the last arriving product will be admitted to service before another

arrival occurs, while if not, the product will have to wait at least

one busy period. These two opposite possibilities increase the value

of the quality’s variance under FIFO regime. When λ approaches µ,the queue becomes very large and every product waits a very long

time, so that V [QF I FO ] decreases sharply.

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Figures 2 further depicts the ratios between the mean qualities of

the two regimes and the ratio between their variances. In figure 2(a)

it is seen that the ratio increases slowly up to λ = 18 and then jumps

to ∞. Figure 2(b) illustrates that the ratio between the variances

increases up to λ = 13.5, then decreases sharply up to λ = 19.2, and

then jumps almost instantly towards ∞. These turning points in

the ration coincides with the two crossing points of Figure 1(b).

(a) Ratio between Expectedquality in LIFO regime vs

FIFO regime(b) Ratio between the qualityVariances of the two regimes

Figure 2: Comparing quality means and variances for FIFO andLIFO service regimes for Exponential service times, when µ = 20,

γ = 1, δ = 1

2.3 Quality analysis for two-phase Erlangservice times

In this section we consider anM/G/1 queue, with Poisson arrival

rate λ and Erlang service time having two consecutive exponential

(µ) durations. That is, B ∼ Eralnд(2, µ). We have B(s) =( µµ+s

)2

,

ρ = 2λµ . The stability condition ρ < 1 translates here to λ <

µ2.

2.3.1 FIFO Service Regime inM(λ)/E(2, µ)/1 queue. Underthe FIFO regime, by using Eq. (8), the mean quality of a departing

product is given by

E[QF I FO ] = δµ(µ − 2λ)

(µ + γ )2 − λ(2µ + γ ). (32)

By using Eq. (10), the variance of QF I FO is given by

V [QF I FO ] = δ22γ µ(µ − 2λ)

(2γ − λ)(µ + 2γ )2 + λµ2−

[δ

γ µ(µ − 2λ)

(γ − λ)(µ + γ )2 + λµ2

]2

.

(33)

2.3.2 LIFO Service Regime inM(λ)/E(2, µ)/1 queue. Under LIFO,by using Eq. (12) the mean quality is given by

E[QLI FO ] = δµ(µ − 2λ)

(µ + γ )2+ δ

λµ2[1 − θ (γ )]

(µ + γ )2(γ + λ[1 − Q(γ )]), (34)

where here θ (γ ) can be calculated by Eq. (4):

θ (γ ) = B(γ + λ − λθ (γ )) =µ2[

µ +[γ + λ − λθ (γ )

] ]2 (35)

This leads to a cubic equation in θ (γ ):

λ2θ3(γ ) − 2λ(µ + γ + λ)θ2(γ ) + (µ + γ + λ)2θ (γ ) − µ2 = 0. (36)

Denote the coefficients in the 3rd degree polynomial given in Eq.

(36) as a = λ2, b = −2λ(µ + γ + λ), c = (µ + γ + λ)2, and d = −µ2.

Then, by using Cardano’s Formula for solving a cubic equation (see

Witula and Slota, 2009 [26]) we get three roots, only one of them is

real and positive, so it is the only valid candidate for θ (γ ):

θ (γ ) = S +T +2(γ + µ + λ)

3λ, (37)

where

S =3

√R +

√P3 + R2 T =

3

√R −

√P3 + R2 , (38)

while,

R =9abc − 27a2d − 2b3

54a3=

2(γ + µ + λ)3 + 27λµ2

54λ3

P =3ac − b2

9a2= −

(γ + µ + λ)2

9λ2

(39)

E[QLI FO ] is calculated for the M(λ)/E(2, µ)/1 queue by substi-

tuting first Eq. (39) in Eq. (38), then Eq. (38) in Eq (37), and finally

Eq. (37) in Eq. (34).

The Variance of QLI FO is given by

V [QLI FO ] =δ2µ(µ − 2λ)

(µ + 2γ )2+ δ2

λµ2[1 − θ (2γ )]

(µ + 2γ )2(2γ + λ[1 − θ (2γ )])−

δ2µ2(µ − 2λ)2

(µ + γ )4+ 2δ2

λµ3(µ − 2λ)[1 − θ (γ )]

(µ + γ )4(γ + λ[1 − θ (2γ )])−

δ2λ2µ4[1 − θ (γ )]2

(µ + γ )4(γ + λ[1 − θ (2γ )])2,

(40)

where θ (2γ ) is the real positive root of the following cubic equation:

λ2θ3(2γ ) − 2λ(µ + 2γ + λ)θ2(2γ ) + (µ + 2γ + λ)2θ (2γ ) − µ2 = 0

(41)

which is solved by Cardano’s formula.

2.3.3 Comparison between LIFO and FIFO inM(λ)/E(2, µ)/1queue. In this sub-section the FIFO and LIFO service regimes are

compared numerically with respect to theM/E2/1 queue.Figures 3 bellow depicts the results for the mean qualities (Figure

3(a)) and variances (Figure 3(b)) for both FIFO (red) and LIFO (blue)

regimes as functions of λ, when µ = 20, γ = 1, δ = 1. E[QLI FO ] is a

concave monotonic decreasing function of λ, and approaches 0.7 asρ → 1, i.e., λ →

µ2= 10. In comparison, E[QF I FO ] is also concave,

but decreases sharply when ρ gets closer to 1, reaching 0 at λ = 10.

Note that E[QLI FO ]/E[QF I FO ] > 1 for all λ.

(a) E[Q ] as a function of λ (b) V [Q ] as a function of λ

Figure 3: Mean and variance of a product’s quality under FIFO andunder LIFO service regimes for two-phase Erlang service times

(E(µ , 2)), as functions of λ, when µ = 20, γ = 1, δ = 12020-03-04 01:10. Page 5 of 1–8.

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Figure 3(b) depicts the behaviour of the variances of the two

service regimes as a function of λ for the same parameter values as

in Figure 3(a). It is seen that both variances increase when λ is small

or moderate. When λ is smaller than 8.8, V [QF I FO ] < V [QLI FO ],

but when λ exceeds λ = 8.8, the ratio turns over and V [QLI FO ] <

V [QF I FO ]. V [QF I FO ] reverses its direction at its highest value

V [QF I FO ] = 0.084 and drops sharply towards 0, crossingV [QLI FO ]

at λ = 9.7.

Figure 4 depicts the ratios between the mean qualities of the two

regimes and the ratio between their variances. In Figure 4(a) the

ratio between the mean qualities increases slowly up to λ = 8.5

and then jumps rapidly to∞. Figure 4(b) illustrates that the ratio

between the variances increases up to λ = 6.3, then decreases

sharply up to λ = 9.4 and finally jumps almost instantly towards

∞. Observe that Figures 4(a) and 4(b) are similar, respectively, to

Figures 2(a) and 2(b).

(a) Ratio between meanqualities:

E[QLI FO ]/E[QF I FO ]

(b) Ratio between the qualityVariances:

E[QLI FO ]/E[QF I FO ]

Figure 4: Comparing quality’s means and variances for FIFO andLIFO service regime for M (λ)/E(2, µ)/1 queue, as a function of λ,

when µ = 20, γ = 1, δ = 1

3 TANDEM JACKSON NETWORK (TJN) WITHTIME-DEPENDENT QUALITYDETERIORATION

Consider an n-site stochastic tandem Jackson network (TJN) where

products arrive to the first site at a Poisson rate λ, and then, after

being processed and inspected at each site, move uni-directionally

from site to site, as depicted in Figure 5 (see Perlman and Yechiali,

2019 [21]).

Figure 5: Tandem Stochastic Network. "Proc(i)" ("Ins(i)") indicatesthe processing (inspection) stage of site i

Each site consists of two separate and independent stages: a

processing stage, followed by an inspection stage, where each stage

is anM/M/1-type queue. A single processing attempt at stage i, Bi ,

is exponentially distributed with mean 1/µi and LST Bi (s) =µi

s+µi ,

and it concludes either successfully, with probability ri , or fails withthe complementary probability 1 − ri . In the latter case, the pro-

cessing operation is immediately repeated with the same success

probability, until a successful attempt. Thus, the total processing

time of a product at site i , Gi , is a Geometric sum of iid Exponen-

tial random variables, and is distributed Exponentially with mean

E(Gi ) =1

µi ri and LST given by G(s) =ri µi

s+ri µi . It is well known (see

Takagi, 1991 [24]) that the output process from an M(λ)/M(∗)/1

queue is also Poisson with the same rate λ, and that the inter-

departure times form a renewal process. Thus, each site can be

treated as an isolatedM/M/1 queue. Upon successful processing

completion at site i, the product moves to the inspection stage of

this site. The inspection time, Di , is exponentially distributed with

mean 1/ξi and LST Di (s) =ξi

s+ξi(i = 1, 2, ...,n).

Upon conclusion of the inspection, the state of the product is

determined according to the following possibilities: (i) with proba-

bility ai the product advances to queue i+1 (1 ≤ i ≤ n−1); (ii) with

probability pi the product is declared to be ’good’ and exits the sys-

tem with a quality value Q depending on the total time it sojourned

in the system; and (iii) with probability fi the product is declaredas ’failed’, discarded and exits the system with quality 0. Clearly,

ai + pi + fi = 1 for every i = 1, 2, ...,n. Thus, the arrival rate to the

processing stage at site i is λi = λ∏i−1

k=1 ak for 1 < i ≤ n− 1, where∏0

k=1 ak = 1. Out of site n, an = 0, and pn + f n = 1. The traffic load

condition for the network to reach stability is λi < min{ri µi , ξi }for each site 1 ≤ i ≤ n.

During its traversal time through the system the product suf-

fers degradation in its quality, depending on the total time it has

sojourned in the system. Let Tj denote a product’s accumulated

traversal time through sites 1 to j . If the product exits the system at

site j (1 ≤ j ≤ n), its quality is either Q = δe−γTj with probability

pj , or zero with probability fj . As indicated above, both stages are

M/M/1 queues. LetWi and Vi denote the total sojourn time in the

processing stage, and in the inspection stage at site i , respectively.Then, the total traversal time of a product through sites 1 to j is

Tj =∑ji=1(Wi +Vi ).

3.1 FIFO Service RegimeThe total sojourn time of a product in an M(λ)/M(µ)/1 queue is

exponentially distributed with mean1

µ−λ . Thus, since the total

processing time in site i is exponentially distributed with parameter

ri µi , and the arrival rate is λi , under the FIFO regime,W F I FOi (a

slight change of notation), the product’s total processing time at

site i , is exponentially distributed with parameter ri µi −λi and LST

W F I FOi (s) =

ri µi−λiri µi−λi+s

.

Similarly, the inspection stage at site i is also an M/M/1 queue

with arrival rate λi and inspection rate ξi . Consequently, the so-journ time V F I FO

i of a product in the inspecting stage there is

also exponentially distributed with parameter ξi − λi and LST

V F I FOi (s) =

ξi−λiξi−λi+s

.

The mean traversal time in the system E[T F I FO ] is given by (see

Perlman and Yechiali [2019])

E[T F I FO ] =

n∑j=1

( ( j−1∏k=1

ak)(pj + fj )

j∑i=1

( 1

ri µi − λi+

1

ξi − λi

) ).

(42)

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The mean traversal time for a product exiting the system as ’good’

is given by (Perlman and Yechiali, 2019 [21])

E[T F I FO |дood] =

∑nj=1

( (∏j−1k=1 ak

)pj

∑ji=1

(1

ri µi−λi+ 1

ξi−λi

) )∑nj=1

( ∏j−1k=1 ak

)pj

(43)

and the mean quality of a product leaving the system is given by

E[QF I FO ] =

n∑j=1

( ( j−1∏k=1

ak)pjδ

j∏i=1

( ri µi − λiri µi − λi + γ

·ξi − λi

ξi − λi + γ

) ),

(44)

where λi = λ∏i−1

k=1 ak .

3.2 LIFO Service RegimeConsidering site i , the LST ofW LI FO

i is given by Eq. (7). Thus, with

ρ = λiri µi and Gi (s) =

ri µiri µi+s , the LST of the the sojourn time in the

processing stage is given by

W LI FOi (s) =

ri µi − λiri µi + s

+λiri µi

[1 − θi (s)

](ri µi + s)(s + λi

[1 − θi (s)

]), (45)

where the LST of θi , the corresponding busy period, is given by

θi (s) =s + λi + ri µi −

√(s + λi + ri µi )2 − 4λiri µ

2λi. (46)

Similarly, the LST of the inspection time in site i , V LI FOi , is

calculated with the aid of Eq. (7). Thus, with ρ = λiξi

and Di (s) =

ξiξi+s

, the LST of V LI FOi is given by

V LI FOi (s) =

ξi − λiξi + s

+λiξi

[1 − ϕi (s)

](ξi + s)(s + λ

[1 − ϕi (s)

]), (47)

where ϕi (s), denoting the busy period in the inspection stage at

site i , is given by

ϕi (s) =s + λi + ξi −

√(s + λi + ξi )2 − 4λiξi2λi

. (48)

Thus, the mean quality of an arbitrary product exiting the system

is given by:

E[QLI FO ] =

n∑i=1

( i−1∏k=1

ak)piE[Q

LI FOi ], (49)

where E[QLI FOi ], the mean quality of a product exiting the system

in site i, is given by:

E[QLI FOi ] = E[−γT

LI FOi ] = δT LI FO

i (γ ) = δi∏

k=1

W LI FOi (γ ) · V LI FO

i (γ )

= δi∏

k=1

(rk µk − λkrk µk + γ

+λkrk µk

[1 − θk (γ )

](rk µk + γ )(γ + λk

[1 − θi (γ )

])

)×(

ξk − λkξk + γ

+λk ξk

[1 − ϕk (γ )

](ξk + γ )(γ + λk

[1 − ϕk (γ )

])

).

(50)

3.3 Comparison between LIFO and FIFO in TJNIn the following section the FIFO and LIFO service regimes are

compared numerically for the tandem Jackson network with n = 2

sites. Two cases are investigated with various parameter values: (i)

identical sites, and (ii) non-identical sites.

3.3.1 Identical sites (n=2). Assume that all processing stages are

statistically identical and all inspection stages are also identical and

compare mean quality as a function of λ. Figure 6 illustrates theresults of E[Q] for various values of ri while γ = 1, δ = 1, µi = 20,

ξi = 20. It is seen that. for all values of ri , LIFO is better than FIFO

for all values of λ. Under both service regimes, as ri increases, themean quality increases due to the decrease in sojourn time at every

station. In addition, for ri = 0.1 (Figure 6a) the shape of E[QLI FO ]

is approximately linear while for ri = 0.5 and ri = 1 E[QLI FO ] is

concave. In all cases E[Q] decreases when λ increases.

(a) E[Q ] for ri = 0.1 (b) E[Q ] for ri = 0.5 (c) E[Q ] for ri = 1

Figure 6: Mean product quality (FIFO and LIFO) in TJN with twonodes as a function of λ for various values of ri while γ = 1, δ = 1,

µi = 20, ξi = 20

Figure 7 illustrates the results of E[Q] as a function of λ for

different values of the shape parameter γ , while δ = 1, ri = 1,

µi = 20, ξi = 20. It is seen that for all values of γ , LIFO is better

than FIFO for all values of λ. When γ = 0.5, for both FIFO and LIFO,

the mean quality is a concave function with increasing difference

between FIFO and LIFO as λ approaches µ. In comparison, forγ = 25

and γ = 50, the mean quality under both regimes is convex, and the

difference between FIFO and LIFO decreases as γ increases. This is

due to the increasing deterioration rate as γ increases.

(a) E[Q ] for γ = 0.5 asa function of λ

(b) E[Q ] for γ = 25 as afunction of λ

(c) E[Q ] forγ = 50 as afunction of λ

Figure 7: Mean product quality for TJN (FIFO and LIFO) with twonodes for various values of γ while δ = 1, ri = 1, µi = 20, ξi = 20

3.3.2 Non-identical processing sites. The parameters for the

first site are: r1 = 1, p1 = 0, a1 = 0.8 and f1 = 0.2, while the

parameters for the second site are: r2 = 1, p2 = 0.2, and f2 = 0.8.

Consider two cases: (i) µ1 = 30, µ2 = 60 and (ii) µ1 = 60, µ2 = 30.

For simplicity we assume that inspection stages are identical with

ξ1 = ξ2 = 20. Figure 8 bellow depicts the value for E[Q] whenδ = 1, and γ = 1. Note that when the processing rate at site 1 is

higher (Fig 8(a)), the deterioration as ρ approaches 1 in both service

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regimes is almost identical, while when the processing rate at site 1

is lower (Fig 8(b)), E[QF I FO ] is bounded (by 0.12) while E[QF I FO ]

is decreasing sharply towards 0, as ρ approaches 1. Those results areexplained by examining the results in previous sections regarding

the M/G/1 queue: when ρ approaches 1, E[QF I FO ] deteriorates

rapidly towards 0.

(a) E[Q ] for µ1 = 60, µ2 = 30 asa function of λ

(b) E[Q ] for µ1 = 30, µ2 = 60 asa function of λ

Figure 8: Mean product quality in a two-node TJN (FIFO and LIFO)with non identical processing stages, while δ = 1, ri = 1, γ = 1,

ξi = 120

4 CONCLUSIONSThis paper analyses the quality of a product traversing through (i) a

single serverM/G/1-type queue, and (ii) a tandem Jackson network.

For theM/G/1 queue, two service time distributions are specifically

studied - Exponential and two-phase Erlang (other service time

distributions are discussed in a technical report by Liberman and

Yechiali, 2019 [17]).

Assuming that the quality of a product deteriorates exponentially

with the time it sojourns in the system, two service disciplines

are compared - FIFO vs. LIFO (the investigation of other service

regimes such as Random Order of Service or Processor Sharing

are left to a further research). Interestingly, it is shown that FIFO

is not necessarily better then LIFO. Numerical results show the

behaviour of the quality mean, as well as the quality variance, as

functions of the various system parameters. It is shown that the

mean quality under LIFO is always greater then FIFO. The first

is bounded below when the traffic intensity approaches 1, while

the latter drops sharply to zero. Similar qualitative results hold

for a tandem Jackson network. A further possible research may

look at concave deteriorating quality function where the product is

considered as perfectly good up to some fixed time threshold.

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